Integral of Reciprocal is Divergent/Unbounded Above/Proof 1

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Theorem

$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$


Proof

From Harmonic Series is Divergent, we have that $\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ diverges to $+\infty$.

Thus from the Cauchy Integral Test:

$\ds \int_1^n \frac {\d x} x \to +\infty$

diverges.

$\blacksquare$


Sources